In Fig. REF , we show the results of GW parameters \(\alpha \)  and \(\beta /H_{\ast }\)  calculated using the gauge-independent potential.
All the samples generate a strong first-order EWPT and survive the experimental constraints.
As illustrated in this figure, there is a considerable portion of samples that can be detected by BBO and DECIGO. However, in contrast with the results obtained from the Landau gauge potential, relatively few sample points fall within the sensitivity of LISA.
This means that the GWs produced from the first-order EWPT in the gauge-independent scalar potential may be too weak to be detected by the LISA interferometer.
A similar conclusion was also found in Ref. [1]}, in which a nonperturbative analysis of the GW power spectrum was adopted to show that GWs produced from a first-order EWPT in a beyond-the-SM scenario described by a SM-like effective theory will be so weak that the LISA experiment is unable to probe.
